CN108280273A - A kind of limited bulk Flow Field Numerical Calculation method based under non equidistance grid analysis - Google Patents

A kind of limited bulk Flow Field Numerical Calculation method based under non equidistance grid analysis Download PDF

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CN108280273A
CN108280273A CN201810014631.7A CN201810014631A CN108280273A CN 108280273 A CN108280273 A CN 108280273A CN 201810014631 A CN201810014631 A CN 201810014631A CN 108280273 A CN108280273 A CN 108280273A
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王镇明
朱君
赵宁
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Nanjing University of Aeronautics and Astronautics
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Abstract

A kind of limited bulk Flow Field Numerical Calculation method based under non equidistance grid analysis, relate to a kind of five rank limited bulk WENO formats that linear power times takes, under the premise of not reducing whole numerical precision, it need to only appoint and take three to add up for the positive number equal to 1 as linear power, then subsequently be calculated.The numeric format that the present invention constructs due to that need not calculate, with mesh scale, weigh under non equidistance grid analysis by related complex linear, so being easier to the Flow Field Calculation being generalized in WENO formats, mobile grid or the adaptive mesh of higher order.In addition, the present invention equally uses stability good with traditional WENO formats, easy to operate and Runge Kutta time discrete methods that are being easily programmed are solved.Finally, the present invention is compared to verify its superiority in Flow Field Calculation with conventional form by specific the numerical example.

Description

Finite volume flow field numerical value calculation method based on non-equidistant grid
Technical Field
The invention discloses a finite volume flow field numerical value calculation method based on a non-equidistant grid, which can be used in the technical fields of calculation mathematics, aerodynamics, magnetohydrodynamics, aerospace, missile launching, automobile industry, civil engineering, environmental engineering, ships, meteorological engineering and the like.
Background
In recent decades, the high-order numerical solution format of the hyperbolic conservation law equation has been the key research content, and the application fields include computational fluid mechanics, computational astronomy, semiconductor simulation, traffic flow models, magnetofluid mechanics and the like. The main difficulty in solving these problems is the complexity of their solutions, which, even if the initial values are smooth, can evolve over time to produce complex solution structures such as shock waves, contact discontinuities and rarefaction waves. In addition, the low-order numerical format can smooth the discontinuities, and the complex solution structures cannot be well distinguished, so that the research on the high-order numerical format aiming at the problems is very necessary. In 1987, Harten and Osher first proposed ENO (essential Non-oscillatoriy) format and successfully applied to the related field. The main idea of the ENO format is to select the smoothest value of the reconstruction target cell among all candidate templates, thereby achieving high accuracy and substantially oscillation-free properties. And because the template selection process exists, the calculation result is wasted in the calculation process, and therefore, the WENO (weighted ENO) format is constructed on the basis of the template selection process. In 1994, the WENO format was first proposed and constructed as a third order finite volume version of the one-dimensional case. In 1996, Jiang and Shu constructed 3 rd and 5 th order finite difference WENO formats and presented a general computational framework for the WENO format, including construction of reconstruction polynomials, calculation of linear weights, calculation of smooth indicators, and calculation of nonlinear weights and final reconstruction values. Based on the work of Jiang and Shu, many subsequent scholars participated in the study of the finite difference/finite volume WENO format, such as: hu, Qiu et al constructs and implements a finite volume WENO format and a central WENO (central WENO) format under unstructured mesh; balsara and Shu et al constructed the finite difference WENO format of the higher order of 7 to 11; qiu proposes HWENO (Hermite WENO) format and the like on the basis of WENO format. In addition, the application fields are very wide, such as: dietrrich, Bernuzzi, Grimm-Strele, Kupka, Leung et al, for applications in astrophysics and geophysics; fierro, Pressel, etc. are used in climate science. Compared with the ENO format, the WENO format performs convex combination on all candidate templates, so that the waste of calculation results is avoided, the property of basically no oscillation of the ENO format is maintained, and the overall precision of the numerical value format is improved.
It should be noted that the main contents of the WENO format are the construction of the reconstruction polynomial, the selection of linear weights, the smoothing indicator, and the calculation of non-linear weights. These calculations are relatively simple in an equidistant grid, and their specific expressions are readily available. But once generalized to a non-equidistant grid, its construction is relatively complex, since its specific expressions are related not only to the physical quantities at the grid points but also to the grid dimensions. In addition, whether the equidistant grid or the non-equidistant grid is adopted, the linear weights in the process must be optimal linear weights to enable the overall format to reach the accuracy of the expected design, otherwise, the accuracy is reduced, and even negative linear weights are easy to occur to cause the format to be unstable. For this reason, Zhu et al designed a finite volume and finite difference WENO format that can have arbitrary linear weights and ensure that the overall accuracy of the format is not lost.
Based on the thought, the invention constructs a finite volume basic weighting oscillation-free format which can be arbitrarily taken by linear weights under non-equidistant grids, and combines 3-order Runge-Kutta time discrete method numerical solution meeting TVD properties to obtain a flow field solution structure in a certain area. In non-equidistant Cartesian grids such as a random grid, a periodic variation grid, a telescopic grid and the like, the flow problem can be solved in a finite volume WENO format which can be arbitrarily selected based on new linear weights of Runge-Kutta time dispersion, and the feasibility of the method is verified through a numerical experiment.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a finite volume flow field numerical calculation method based on a non-equidistant grid, which can carry out numerical calculation simulation on the flow problem in the flow field by utilizing a high-precision calculation method under a common non-equidistant Cartesian grid.
In order to achieve the purpose, the invention adopts the following technical scheme:
a finite volume flow field numerical calculation method based on a non-equidistant grid is characterized in that a basic weighting non-oscillation format with 5-order precision is constructed in a non-equidistant non-uniform Cartesian grid to carry out numerical calculation simulation in a flow field, and the method specifically comprises the following steps:
step 1, generating a non-equidistant Cartesian coordinate system in a flow field numerical calculation area;
and 2, constructing a finite volume WENO format in the established non-equidistant Cartesian coordinate system, wherein the linear weight can be taken at will and is independent of the grid scale and type, the 5-order precision of the format can be kept, and then solving a control equation set of the inviscid fluid by combining a Runge-Kutta time dispersion method to obtain high-precision numerical value approximate values of physical quantities of all non-equidistant grid unit points in the flow field.
In order to optimize the technical scheme, the specific measures adopted further comprise:
in the step 1, the non-equidistant cartesian coordinate system specifically comprises:
random gridding:
Δxi＝0.5Δx+(1.5Δx-0.5Δx)·random_number(i)，
Δyj＝0.5Δy+(1.5Δy-0.5Δy)·random_number(j)，
wherein the subscripts i, j denote grid numbers in the x and y directions, respectively, Δ xiDenotes the step size, Δ y, corresponding to the ith grid in the x-directionjRepresenting the step length corresponding to the jth grid in the y direction, Δ x representing the average step length in the x direction, Δ y representing the average step length in the y direction, random _ number (i) and random _ number (j) representing random numbers between 0 and 1 corresponding to the ith and jth grids, respectively;
or a periodically varying grid:
wherein L isx，LyRespectively representing the lengths of the calculation regions in the x-direction and y-direction, Nx，NyRespectively representing the number of grid units in the x direction and the y direction of the calculation area;
or a telescopic grid:
Δxi＝Δxi-1·α，
Δyj＝Δyj-1·β，
here, α and β represent the expansion ratio in the x direction and the y direction, respectively.
The step 2 specifically comprises:
2.1 integrating two sides of a control equation set of the inviscid fluid in a target unit at the same time to construct a spatial derivative term in a finite volume WENO format so as to obtain a semi-discrete finite volume format;
2.2 solving the semi-discrete finite volume format by using a Runge-Kutta time discrete method, thereby obtaining a space-time full-discrete finite volume format of a control equation set, wherein a group of nonlinear weights can be arbitrarily valued and are independent of grid scale and type, and only positive numbers with sum of 1 are required;
2.3 the space-time fully-discrete finite volume format is an iterative formula in the time direction, the related physical quantity of the initial state of the flow field is given, and the high-precision approximate value of the physical quantity of the flow field at a certain moment or in a stable state can be obtained by continuously circulating according to the iterative formula.
In step 2, the control equation set of the viscous-free fluid is as follows:
where t denotes a time variable, x, y denotes a space variable, and U ═ p, ρ U, ρ v, E)TDenotes a conservation variable, f (u) ═ p u, p u2+p，ρuv，u(E+p))T，g(U)＝(ρv，ρuv，ρv2+p，v(E+p))TF (U), g (U) denotes flux, f (U)xDenotes f (U) derivative of x, g (U)yG (U) is derived from y, and ρ, p, u, v, and E are the fluid density,Pressure, horizontal velocity, vertical velocity and energy, T denotes transposition, U0Indicating the initial state value.
The step 2.1 specifically comprises:
for both sides of the control equation to be in the target grid cell Ii，jThe internal integral has:
wherein,representing the conservation variable U (x, y, t) in the target unit Ii，jThe average value of the values of (a) to (b),meaning that the derivative is taken over a time variable t,indicating the corresponding integrand phi in the corresponding integration interval [ a, b ]]Internal integration;
constructing a spatial derivative term in a finite volume WENO format by the following specific process:
2.1.1 solving the integral term in the right hand end of the above equation using the three-point Gauss product equation, i.e.
Wherein G ismExpressing the product-solving nodes corresponding to the three-point Gauss product-solving formula;
2.1.2 Replacing the throughput of each product node in the Gauss product equation with the Lax-Friedrichs numerical flux approximation, i.e.
Wherein,representing corresponding pointsThe high-order approximation on the right-hand side,representing corresponding pointsThe left-hand high-order approximation,representing corresponding pointsAn upper-side high-order approximation,representing corresponding pointsUpper-side higher-order approximation, α, represents the maximum value of the flux f (U) or g (U) derivative of the conservative variable U;
2.1.3 high order approximation under non-equidistant griddingThe higher order approximation of other points is similarly determined, in particularThe following were used:
2.1.3.1 fix the spatial variable y, i.e. assume y ═ ykUsing grid meanTo obtainThe procedure of the higher order approximation of (a) is as follows:
a) selecting a master templateThereby obtaining a fourth order reconstruction polynomial p1(x，yk) Which satisfies the following conditions:
the specific expression is obtained as follows:
the coefficient of the linear equation satisfies the linear equation set U ═ C × A, and the specific form is as follows:
A＝(a1，a2，a3，a4，a5)T，
wherein U, A represents vector, C represents coefficient matrix, T represents transpose to vector or matrix;
b) selecting two small templatesThereby obtaining two linear reconstruction polynomials p2(x，yk)，p3(x，yk) Which respectively satisfy:
the specific expression is as follows:
c) arbitrarily selecting three positive numbers smaller than 1 and with a sum of 1 as the linear weight gamman(n＝1，2，3)；
d) Calculate each templateCorresponding smooth indicator βn，k(n ═ 1, 2, 3), the calculation formula is as follows:
where n denotes the order of the corresponding polynomial, α denotes a summation index, r denotes the degree of the corresponding polynomial,representing a polynomial pn(x，yk) α times partial derivatives of the independent variable x;
the specific expression corresponding to the smoothness indicator is as follows:
e) according to a linear weight gammanAnd a smoothness indicator βn，kSolving for a non-linear weight ωn，kThe formula is as follows:
wherein,is a transition value in the calculation process;
f) to obtainThe reconstruction approximation expression of (1):
2.1.3.2 let y be yk-2，yk-1，yk+1，yk+2And repeating the step 2.1.3.1 to obtain
A value of (d);
2.1.3.3 space variable x is fixed, i.e. x is xi+1/2And x ═ xi-1/2Using the results of step 2.1.3.1 and step 2.1.3.2Value of obtainingThe final higher order approximation of (c):
2.1.3.4 repeating steps 2.1.3.1 through 2.1.3.3 to yield similarlyA value of (d);
2.1.4 substituting the value of step 2.1.3 into step 2.1.2, and then substituting into step 2.1.1, a semi-discrete finite volume format of the fluid control equation set can be obtained and expressed as:
in said step 2.2, it is assumed that the time step is Δ t, tnIt is shown that the n-th temporal layer,to representAt t, atnThe time layer is obtained by the process of step 2.1Then theDispersing the semi-discrete finite volume format by using a Runge-Kutta time dispersion method satisfying the TVD property to obtain a space-time full-discrete finite volume format, wherein the method comprises the following steps:
wherein,andto calculate intermediate transition values.
In the step 2.3, the space-time full-discrete finite volume format obtained after the dispersion in the space direction and the time direction is completed is an iterative formula related to the time layer, and the high-precision approximate value of the physical quantity at a certain moment or in a stable state of the flow field can be obtained by continuously circulating according to the iterative formula after the values of the relevant physical quantity at the initial state of the flow field are known.
The invention has the beneficial effects that: a new finite volume WENO format with linear weight with 5-order precision is constructed in non-equidistant Cartesian grids such as a random grid, a periodic variation grid, a telescopic grid and the like, and the flow problem of complex solution structures including shock waves, contact discontinuities, sparse waves and the like in a flow field is numerically simulated by combining a Runge-Kutta time dispersion method, and the method has the advantages that: 1) on the premise of ensuring the overall accuracy of the format, the linear weight in the construction process can be taken as any value, is irrelevant to the grid scale, and only needs to satisfy the condition that the sum is greater than 0 and equal to 1; 2) avoiding handling negative non-linear weights present in traditional finite volume WENO formats; 3) the fluid flow problem can be treated in various non-equidistant grids; 4) the method can be more simply popularized to the high-dimensional problem or the problem under the mobile grid.
The traditional finite volume WENO numerical method has the advantages that the linear weight in the non-equidistant grid can reach the expected design precision only by counting the optimal linear weight, the calculation is related to the grid type and the grid scale, the calculation is relatively complex, particularly, the calculation of the linear weight corresponding to each Gauss integration node is needed when the method is popularized to a high-dimensional situation, the calculation is more complex, and the negative linear weight is easy to occur, so that the format is unstable.
Drawings
FIG. 1 is a schematic diagram of three non-equidistant grids.
Fig. 2 is a pressure contour plot of the dual mach problem computed in the random grid and periodic grid in example two.
Fig. 3 is a density contour plot of the dual mach problem computed in the random grid and periodic grid in example two.
Fig. 4 is a density contour plot of the symmetric Riemann problem computed in the random grid and the periodic grid in the third example.
FIG. 5 is a schematic grid diagram illustrating the calculation of the Rayleigh-Taylor instability problem in the fourth embodiment.
FIG. 6 is a density contour plot of the Rayleigh-Taylor instability problem calculated using any of the three linear weights in example four.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
First, a new finite volume WENO format under a non-equidistant grid is constructed.
Consider the two-dimensional flow field governing equation:
where t denotes a time variable, x, y denotes a space variable, U ═ p, ρ U, ρ v, E)TDenotes a conservation variable, f (u) ═ p u, p u2+p，ρuv，u(E+p))T，g(U)＝(ρv，ρuv，ρv2+p，v(E+p))TF (U), g (U) denotes flux, f (U)xDenotes f (U) derivative of x, g (U)yG (U) is derived from y, rho, p, U, v, E respectively represent fluid density, pressure, horizontal velocity, vertical velocity and energy, T represents transposition, U0Indicating the initial state value. It is to be noted that all the physical quantities mentioned above are functions with respect to time and space variables, i.e. changes with time and space position.
Let xi+1/2-xi-1/2＝Δxi，yj+1/2-yj-1/2＝ΔyjThe center of the grid isGrid cell is Ii，j＝[xi-1/2，xi+1/2]×[yj-1/2，yj+1/2]And subscripts i and j are coordinate serial numbers. For convenience of explanation, the present invention takes three non-equidistant grids of a random grid, a periodically varying grid and a scaling grid, as shown in fig. 1.
Definition ofIndicating U in grid cell Ii，jInner average value, i.e.tnDenotes the nth time layer, Δ t denotes the time step, then tn+1＝tn+Δt。To representAt the value of the nth time layer, the following processes are all performed at the nth time layer, so the time variable t is temporarily not considered, i.e., the time variable t is about to beIs abbreviated as
Step 1, high-order approximation under non-equidistant gridValue of (A), GmA representation of the Gauss node is shown,representing corresponding pointsThe high-order approximation on the right-hand side,representing corresponding pointsLeft high order approximation, specifically as follows:
step 1.1, fix the spatial variable y, assuming y ═ ykUsing grid meanTo obtainThe procedure of the higher order approximation of (a) is as follows:
in the step 1.1.1, the method comprises the following steps of,selecting a master templateThereby obtaining a fourth order reconstruction polynomial p1(x，yk) Which satisfies the following conditions:
obtain a specific expression of
Wherein a is1，a2，a3，a4，a5The linear equation set U is C multiplied by A, and the concrete form is as follows:
A＝(a1，a2，a3，a4，a5)T， (5)
step 1.1.2, two small templates are selectedThereby obtaining two linear reconstruction polynomials p2(x，yk)，p3(x，yk) Which respectively satisfy:
the specific expression is
Step 1.1.3, arbitrarily selecting three positive numbers less than 1 (the sum is 1) as the linear weight gamman(n is 1, 2, 3), the conventional 5-order finite volume WENO format must be the optimal linear weight here, and the calculation is relatively complicated in relation to the grid scale and the corresponding target point, while the present invention takes the same set of numbers for simplicity in numerical calculation.
Step 1.1.4, calculate each templateCorresponding smooth indicator βn，k(n ═ 1, 2, 3), the calculation formula is as follows:
where n denotes the order of the corresponding polynomial, α denotes a summation index, r denotes the degree of the corresponding polynomial,representing a polynomial pn(x，yk) The argument x is subjected to α partial derivatives.
The specific expression corresponding to the smoothness indicator is
Step 1.1.5, according to the linear weight gammanAnd a smoothness indicator βn，kSolving for a non-linear weight ωn，kThe formula is as follows:
wherein,for calculating the transition value in the process, epsilon is a very small number, the denominator is prevented from being 0, and the invention takes 10 values-6。
Step 1.1.6, finally, obtainingThe reconstruction approximation expression of (1):
step 1.2, then, let y ═ y, respectivelyk-2，yk-1，yk+1，yk+2Repeating the step 1.1 by only changing subscripts k in the formulas (2) to (16) into k-2, k-1, k +1, k +2, respectively, thereby obtaining the compound
The value of (c).
Step 1.3, fix the space variable x, let x ═ x respectivelyi+1/2And x ═ xi-1/2Using the products obtained in step 1.1 and step 1.2And (4) repeating the step 1.1, and only replacing the corresponding values in the step 1.1 with the five values to obtain the valueThe final higher order approximation of (c):
step 2, high-order approximation under non-equidistant gridValue of (A), GmA representation of the Gauss node is shown,representing corresponding pointsAn upper-side high-order approximation,representing corresponding pointsUpper order approximation. Analogously to step 1, the spatial variable x is first fixed as xlUsing grid meanTo obtainA high order approximation of (d); secondly, let x be x respectivelyl-2，xl-1，xl+1，xl+2Are respectively obtained A value of (d); further, a space variable y is fixed, and y is made equal to yj-1/2And y ═ yj+1/2Repeating the above steps by using the above five values to obtainThe final higher order approximation of (c):
step 3, the products obtained in the step 1 and the step 2Andsubstituting the value into the following Lax-Friedrichs numerical flux approximation formula to replace the real flux corresponding to each product node in the Gauss product calculation formula, namely
Where α represents the maximum value of the derivative of the flux f (U) or g (U) on the conservative variable U.
And 4, further obtaining the integral of the flux in the corresponding interval (namely the flux of the corresponding edge) by a three-point Gauss product-solving formula, namely substituting the following formulas (17) and (18):
wherein G ismRepresents the product nodes corresponding to the three-point Gauss product formula, which are respectively0；αmThe product coefficient corresponding to each node of the three-point Gauss product formula is shown as
Step 5, simultaneously arranging the two sides of the control equation (1) in the target grid unit Ii，jThe internal integral has:
and 6, substituting the formulas (19) and (20) into the formula (21) to obtain a high-order approximate value of a right-end term of the formula (21), and expressing the high-order approximate value as the right-end termThen a semi-discrete finite volume format is obtained containing the time derivative term:
step 7, the formula (24) is an ordinary differential equation about a time variable t, and is solved by a three-order Runge-Kutta method with TVD property, as follows:
wherein,andto calculate intermediate transition values.
And 8, combining the expressions (24) and (25) to obtain a space-time fully discrete format, namely an iterative formula about time t, and assuming that the initial state of the physical quantity of the flow field is known, performing cyclic iteration to obtain the value of each physical quantity of the flow field at a certain moment or stably, so as to achieve the purpose of numerically simulating the flow of the flow field.
Four examples are given below as specific examples of the disclosed method.
Embodiment one, precision test problem. Considering equation set (1), the initial conditions are: ρ (x, y, 0) is 1+0.2sin (pi (x + y)), u (x, y, 0) is 0.7, v (x, y, 0) is 0.3, and p (x, y, 0) is 1. The exact solution is: ρ (x, y, t) is 1+0.2sin (pi (x + y-t)), u (x, y, t) is 0.7, v (x, y, t) is 0.3, and p (x, y, t) is 1. The calculation termination time is t-2. The calculation area is (x, y) belongs to [0, 2 ]]×[0，2]. In order to show that the linear weight in the numerical calculation method can be arbitrarily valued, the method only takes five groups of numbers with different laws as the linear weight: (1) gamma ray1＝0.98，γ2＝0.01，γ3＝0.01；(2)γ1＝0.495，γ2＝0.495，γ3＝0.01；(3)γ1＝1/3，γ2＝1/3，γ3＝1/3；(4)γ1＝0.001，γ2＝0.001，γ3＝0.998；(5)γ1＝0.3，γ2＝0.4，γ3＝0.3。
TABLE 1 comparison of numerical precision of any five linear weights with conventional WENO format
Example two, the double mach reflex problem. As shown in fig. 2 and 3, a strong shock wave with mach number 10 is incident at an included angle of 60 ° with the x-axis at a position separated from the left boundary 1/6, and the left and right state values are: (ρ)L，uL，vL，EL)＝(8，7.145，-4.125，563.544)，(ρR，uR，vR，ER) (1.4, 0, 0, 2.5). The CFL count is 0.6, the computational grid is 300 × 100, and the termination time t is 0.2.
Example three, radial symmetry Riemann problem. As shown in fig. 4, taking into account the fluid mechanics equation set (1), the initial conditions are:
the calculation region is [0, 1] × [0, 1], the CFL number is 0.6, the calculation grid is 100 × 100, and the calculation termination time t is 0.13.
Example four, the Rayleigh-Taylor instability problem. As shown in fig. 5 and 6, this problem is considered that the interface of two light and heavy fluids is unstable under the action of gravity or inertia force, so that turbulent mixing occurs, and a very complicated flow field structure is generated. Considering equation (1) with the gravity source term, assuming the gravity direction is upward, take the calculation area to be [0, 0.25 ]]×[0，1.0]. The initial interface is located at y-0.5, the heavy fluid is located below the interface, and the initial state is: ρ ═ 2.0, u ═ 0.0, v ═ 0.025C · cos (8 π x), p ═ 2y + 1; the light fluid is located above the interface, and the initial state is: ρ -1.0, u-0.0, v-0.025C · cos (8 π x),wherein C is the speed of sound,5/3. The left and right boundaries are reflection boundary conditions, the upper and lower boundaries are no-reflection boundary conditions, the upper boundary parameter is ρ 1.0, u is 0.0, v is 0.0, p is 2.5, the lower boundary parameter is ρ 2.0, u is 0.0, v is 0.0, and p is 1.0. The calculation termination time t is 1.95.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (7)

1. A finite volume flow field numerical calculation method based on a non-equidistant grid is characterized in that a basic weighting non-oscillation format with 5-order precision is constructed in a non-equidistant non-uniform Cartesian grid to carry out numerical calculation simulation in a flow field, and the method specifically comprises the following steps:
step 1, generating a non-equidistant Cartesian coordinate system in a flow field numerical calculation area;
and 2, constructing a finite volume WENO format in the established non-equidistant Cartesian coordinate system, wherein the linear weight can be taken at will and is independent of the grid scale and type, the five-order precision of the format can be kept, and then solving a control equation set of the inviscid fluid by combining a Runge-Kutta time dispersion method to obtain high-precision numerical value approximate values of physical quantities of all non-equidistant grid unit points in the flow field.
2. The finite volume flow field numerical computation method based on non-equidistant grids as claimed in claim 1, wherein: in the step 1, the non-equidistant cartesian coordinate system specifically comprises:
random gridding:
Δxi＝0.5Δx+(1.5Δx-0.5Δx)·random_number(i)，
Δyj，＝0.5Δy+(1.5Δy-0.5Δy)·random_number(j)，
wherein subscripts t, j denote grid numbers in x and y directions, respectively, Δ xiDenotes the step size, Δ y, corresponding to the ith grid in the x-directionjRepresenting the step length corresponding to the jth grid in the y direction, Δ x representing the average step length in the x direction, Δ y representing the average step length in the y direction, random _ number (i) and random _ number (j) representing random numbers between 0 and 1 corresponding to the ith and jth grids, respectively;
or a periodically varying grid:
wherein L isx，LyRespectively representing the lengths of the calculation regions in the x-direction and y-direction, Nx，NyRespectively representing the number of grid units in the x direction and the y direction of the calculation area;
or a telescopic grid:
Δxi＝Δxi-1·α，
Δyj＝Δyj-1·β，
here, α and β represent the expansion ratio in the x direction and the y direction, respectively.
3. The finite volume flow field numerical computation method based on non-equidistant grids as claimed in claim 2, wherein: the step 2 specifically comprises:
2.1 integrating two sides of a control equation set of the inviscid fluid in a target unit at the same time to construct a spatial derivative term in a finite volume WENO format so as to obtain a semi-discrete finite volume format;
2.2 solving the semi-discrete finite volume format by using a Runge-Kutta time discrete method, thereby obtaining a space-time full-discrete finite volume format of a control equation set, wherein a group of nonlinear weights can be arbitrarily valued and are independent of grid scale and type, and only positive numbers with sum of 1 are required;
2.3 the space-time fully-discrete finite volume format is an iterative formula in the time direction, the related physical quantity of the initial state of the flow field is given, and the high-precision approximate value of the physical quantity of the flow field at a certain moment or in a stable state can be obtained by continuously circulating according to the iterative formula.
4. A finite volume flow field numerical computation method based on non-equidistant grid as claimed in claim 3, wherein: in step 2, the control equation set of the viscous-free fluid is as follows:
where t denotes a time variable, x, y denotes a space variable, and U ═ p, ρ U, ρ v, E)TDenotes a conservation variable, f (u) ═ p u, p u2+p，ρuv，u(E+p))T，g(U)＝(ρv，ρuv，ρv2+p，v(E+p))TF (U), g (U) denotes flux, f (U)xDenotes f (U) derivative of x, g (U)yG (U) is derived from y, rho, p, U, v, E respectively represent fluid density, pressure, horizontal velocity, vertical velocity and energy, T represents transposition, U0Indicating the initial state value.
5. The finite volume flow field numerical computation method based on non-equidistant grids as claimed in claim 4, wherein: the step 2.1 specifically comprises:
for both sides of the control equation to be in the target grid cell Ii，jThe internal integral has:
wherein,representing the conservation variable U (x, y, t) in the target unit Ii，jThe average value of the values of (a) to (b),meaning that the derivative is taken over a time variable t,indicating the corresponding integrand phi in the corresponding integration interval [ a, b ]]Internal integration;
constructing a spatial derivative term in a finite volume WENO format by the following specific process:
2.1.1 solving the integral term in the right hand end of the above equation using the three-point Gauss product equation, i.e.
Wherein G ismExpressing the product-solving nodes corresponding to the three-point Gauss product-solving formula;
2.1.2 Replacing the throughput of each product node in the Gauss product equation with the Lax-Friedrichs numerical flux approximation, i.e.
Wherein,representing corresponding pointsThe high-order approximation on the right-hand side,representing corresponding pointsThe left-hand high-order approximation,representing corresponding pointsAn upper-side high-order approximation,representing corresponding pointsUpper-side higher-order approximation, α, represents the maximum value of the flux f (U) or g (U) derivative of the conservative variable U;
2.1.3 high order approximation under non-equidistant griddingThe higher order approximation of other points is similarly calculated as follows:
2.1.3.1 fix the spatial variable y, i.e. assume y ═ ykBy means of netsMean value of squaresTo obtainThe procedure of the higher order approximation of (a) is as follows:
a) selecting a master templateThereby obtaining a fourth order reconstruction polynomial p1(x，yk) Which satisfies the following conditions:
the specific expression is obtained as follows:
the coefficient of the linear equation satisfies the linear equation set U ═ C × A, and the specific form is as follows:
A＝(a1，a2，a3，a4，a5)T，
wherein U, A represents vector, C represents coefficient matrix, T represents transpose to vector or matrix;
b) selecting two small templatesThereby obtaining two linear reconstruction polynomials p2(x，yk)，p3(x，yk) Which respectively satisfy:
the specific expression is as follows:
c) arbitrarily selecting three positive numbers smaller than 1 and with a sum of 1 as the linear weight gamman(n＝1，2，3)；
d) Calculate each templateCorresponding smooth indicator βn，k(n ═ 1, 2, 3), the calculation formula is as follows:
where n denotes the order of the corresponding polynomial, α denotes a summation index, r denotes the degree of the corresponding polynomial,representing a polynomial pn(x，yk) α times partial derivatives of the independent variable x;
the specific expression corresponding to the smoothness indicator is as follows:
e) according to a linear weight gammanAnd a smoothness indicator βn，kSolving for a non-linear weight ωn，kThe formula is as follows:
wherein,is a transition value in the calculation process;
f) to obtainThe reconstruction approximation expression of (1):
2.1.3.2 let y be yk-2，yk-1，yk+1，yk+2And repeating the step 2.1.3.1 to obtain
A value of (d);
2.1.3.3 space variable x is fixed, i.e. x is xi+1/2And x ═ xi-1/2Using the results of step 2.1.3.1 and step 2.1.3.2Value of obtainingFinal higher order approximation ofExpression:
2.1.3.4 repeating steps 2.1.3.1 through 2.1.3.3 to yield similarlyA value of (d);
2.1.4 substituting the value of step 2.1.3 into step 2.1.2, and then substituting into step 2.1.1, a semi-discrete finite volume format of the fluid control equation set can be obtained and expressed as:
6. the finite volume flow field numerical computation method based on non-equidistant grids as claimed in claim 5, wherein: in said step 2.2, it is assumed that the time step is Δ t, tnIt is shown that the n-th temporal layer,to representAt t, atnThe time layer is obtained by the process of step 2.1Then, a half-discrete finite volume format is discretized by a Runge-Kutta time discrete method meeting the TVD property to obtain a space-time full-discrete finite volume format, which comprises the following steps:
wherein,andto calculate intermediate transition values.
7. The finite volume flow field numerical computation method based on non-equidistant grids as claimed in claim 6, wherein: in the step 2.3, the space-time full-discrete finite volume format obtained after the dispersion in the space direction and the time direction is completed is an iterative formula related to the time layer, and the high-precision approximate value of the physical quantity at a certain moment or in a stable state of the flow field can be obtained by continuously circulating according to the iterative formula after the values of the relevant physical quantity at the initial state of the flow field are known.
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